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We present an algorithm to accelerate a large class of image processing operators. Given a low-resolution reference input and output pair, we model the operator by fitting local curves that map the input to the output. We can then produce a full-resolution output by evaluating these low-resolution curves on the full-resolution input. We demonstrate that this faithfully models state-of-the-art operators for tone mapping, style transfer, and recoloring. The curves are computed by lifting the input into a bilateral grid and then solving for the 3D array of affine matrices that best maps input color to output color per x, y , intensity bin. We enforce a smoothness term on the matrices which prevents false edges and noise amplification. We can either globally optimize this energy, or quickly approximate a solution by locally fitting matrices and then enforcing smoothness by blurring in grid space. This latter option reduces to joint bilateral upsampling [Kopf et al. 2007] or the guided filter [He et al. 2013], depending on the choice of parameters. The cost of running the algorithm is reduced to the cost of running the original algorithm at greatly reduced resolution, as fitting the curves takes about 10 ms on mobile devices, and 1--2 ms on desktop CPUs, and evaluating the curves can be done with a simple GPU shader.